Optimizing chaotic systems by orbit counting and Fourier spectrum: FPGA implementation and image encryption application

The optimization of chaotic systems has been performed by considering dynamical characteristics of the mathematical models. The proposed work shows the application of genetic algorithms (GAs) to optimize the chaotic behavior of three well-known systems, namely: Lorenz, Chen and Lü. The parameters of the chaotic systems are varied in a specific range of values considered as the search space, and the evaluation of the mathematical model is performed by applying the Forward Euler method. The contribution presented herein is that the chaotic behavior is evaluated by counting the orbits in an attractor and the sparsity of them. In addition, the chaotic behavior is guaranteed by evaluating the Fourier spectrum of the time series. The solutions provided by the GA, are then implemented on a field-programmable gate array (FPGA) to verify the experimental generation of chaotic attractors. Finally, two optimized chaotic systems are synchronized and used to encrypt an image, thus confirming the appropriateness of optimizing the chaotic behavior by orbit counting and Fourier spectrum analysis.